Weak, Strong, and Linear Convergence of a Double-Layer Fixed Point Algorithm
研究实希尔伯特空间中有限族集合的一致凸可行性问题，提出一种双层不动点算法，通过内外两层控制实现弱、强或线性收敛，并数值验证其加速效果。
In this article we consider a consistent convex feasibility problem in a real Hilbert space defined by a finite family of sets $C_i$. We are interested, in particular, in the case where for each $i$, $C_i={Fix} U_i=\{z\in \mathcal H\mid p_i(z)=0\}$, $U_i\colon\mathcal H\rightarrow \mathcal H$ is a cutter and $p_i\colon\mathcal H\rightarrow [0,\infty)$ is a proximity function. Moreover, we make the following assumption: the computation of $p_i$ is at most as difficult as the evaluation of $U_i$ and this is at most as difficult as projecting onto $C_i$. We study a double-layer fixed point algorithm which applies two types of controls in every iteration step. The first one---the outer control---is assumed to be almost cyclic. The second one---the inner control---determines the most important sets from those offered by the first one. The selection is made in terms of proximity functions. The convergence results presented in this manuscript depend on the conditions which, first, bind together the sets, the operators, and the proximity functions and, second, connect the inner and outer controls. In particular, weak regularity (demi-closedness principle), bounded regularity, and bounded linear regularity imply weak, strong, and linear convergence of our algorithm, respectively. The framework presented in this paper covers many known (subgradient) projection algorithms already existing in the literature, for example, those applied with (almost) cyclic, remotest-set, maximum displacement, most-violated constraint, and simultaneous controls. In addition, we provide several new examples, where the double-layer approach indeed accelerates the convergence speed as we demonstrate numerically.